The proof of Theorem REF  shares some basic ideas with that for the complete data case (See, e.g. Theorem 2 in Cai et al. [1]}). However, it relies on a new key technical tool which is a large deviation result for a sub-matrix of the generalized sample covariance matrix under the spectral norm. This random matrix result for the case of missing data, stated in the following lemma, can be potentially useful for other, related high-dimensional missing data problems.
The proof of Lemma REF , given in Section , is more involved than the complete data case, as in the generalized sample covariance matrix each entry, \(\hat{\sigma }_{ij}^\ast \) , is the average of a varying number of samples.
